9.4.5 Illustrating the Fundamental Theorem of Calculus

Question 1

Illustrating the Fundamental Theorem of Calculus

Answer 1

• The fundamental theorem of calculus enables you to evaluate definite integrals, thereby finding the area between the x-axis and a curve that lies above it. Area =
• When the limits of integration are not given by the problem, find them by determining where the curve intersects the x-axis.
• The fundamental theorem of calculus states that if f is continuous on [a, b] and F is an antiderivative of f on that interval, then

Question 2

note

Answer 2

• The fundamental theorem of calculus implies that you can calculate the area between the x-axis and a curve that lies above it by finding the antiderivative of the function and evaluating it at the specified endpoints.
• You can calculate the area of the region under such a curve without graphing it. However, it is a good idea to draw a picture to make sure your answer makes sense.
• Sometimes the endpoints may not be specified from the outset.
• You want the area of the region below the curve and above the x-axis. The function is a quadratic, so the curve will be parabolic. The negative coefficient of x-squared means that the parabola will open downward.
• First determine if the curve crosses the x-axis. The equation of the x-axis is y = 0, so set the function equal to 0 and solve for x.
• There are two solutions, which correspond to the
  x-intercepts of the parabola. They are the endpoints of the region. Use them as the limits of integration of the definite integral of f(x) to determine the area of the region.

Question 3

Evaluate the integral using the fundamental theorem of calculus.−2∫−3(4x+3)dx

Answer 3

−7

Question 4

What is the area bound by the curves y=−1/x+1/x^2, x=−3, x=−1, and the x-axis?

Answer 4

2/3+ln3

Question 5

Which of the following expressions represents the area of the shaded region in this diagram?

Answer 5

1∫−1(2x^3−x+3)dx

Question 6

Consider the function y = −x^ 3 + 5x ^2 − 6x. What is the area of the region bound by the curve and the x‑axis in the first quadrant?

Answer 6

5/12

Question 7

What is the area bound between the curve y = |x|^1/2 and the x‑axis between x = −3 and x = 2?

Answer 7

4√2/3+2√3

Question 8

Consider the function y = e ^x. What is the area bound between this curve and the line e ^x = 4 in the first quadrant?

Answer 8

3

Question 9

Find the area under the function y=−x^2+x+6 and above the x-axis.

Answer 9

20 5/6

Question 10

Find the area bound between the curve

y = 2x^ 1/3 and the x‑axis on the interval [1, 8].

Answer 10

22 1/2